Chargement des Librairies Nécessaires

#install.packages('readxl')
library(readxl)

#install.packages('tidyverse')
library(tidyverse)

#install.packages("fpp2")
library(fpp2)

#install.packages("forecast")
library(forecast)

#install.packages("ggplot2")
library(ggplot2)

install.packages("openxlsx")
library(openxlsx)

Importation des Données

data <- read_excel('/Users/annabellenarsama/Desktop/SeriesTemporelles/electrain.xlsx')
print(data)

Pour des raisons de symétrie, nous enlevons les 92 premières lignes qui correspondent au premier jour du mois, mais auxquelles il manque les 4 premières valeurs.

newdata <- data[-(1:91), ]
print(newdata)

Transformation des Données en Série Temporelle

newdata["jour"] <- weekdays(as.POSIXct(newdata$Timestamp, format="%m/%d/%Y %H:%M"))
newdata["heure"] <- format(strptime(newdata$Timestamp, format="%m/%d/%Y %H:%M"))
print(newdata)

elec <- ts(newdata$`Power (kW)`, start=c(1,6), end=c(51,96), freq=96)
print(elec)

Time Series:
Start = c(1, 6) 
End = c(51, 96) 
Frequency = 96 
   [1] 163.1 154.4 152.2 158.7 163.8 158.7 152.3 155.2 155.9 152.1 154.1
  [12] 155.9 156.8 153.9 152.2 165.6 168.8 160.5 160.6 161.1 160.7 157.0
  [23] 161.7 158.4 165.8 166.4 168.0 159.5 164.2 170.3 178.8 181.5 182.4
  [34] 270.9 269.4 273.1 268.3 277.7 269.8 268.0 258.3 260.7 257.2 256.3
  [45] 255.0 270.9 269.6 307.6 283.5 266.1 295.9 278.5 269.5 297.5 294.4
  [56] 300.7 287.5 288.4 288.4 283.9 310.4 285.3 288.9 277.1 280.3 271.8
  [67] 301.8 282.2 285.0 326.2 311.7 313.4 305.0 300.7 303.2 299.8 307.7
  [78] 315.8 304.1 305.6 290.9 289.4 285.2 276.6 282.0 291.6 286.9 290.1
  [89] 285.1 285.7 286.6 281.9 281.3 190.3 194.8 163.7 159.3 158.4 152.9
 [100] 159.6 161.6 159.9 149.8 148.7 153.2 150.6 151.6 160.6 165.7 154.9
 [111] 149.0 163.5 161.2 158.6 154.9 162.0 162.7 160.3 162.3 168.3 165.3
 [122] 166.8 166.7 172.6 171.1 175.5 185.4 188.8 186.6 267.0 267.3 262.6
 [133] 258.2 262.0 259.4 256.6 258.2 262.8 264.7 262.9 263.8 270.7 264.2
 [144] 269.3 265.9 265.7 261.7 270.9 280.0 281.0 278.9 276.5 276.0 273.1
 [155] 270.2 276.3 265.0 266.2 267.3 275.4 273.0 273.0 272.5 266.3 284.0
 [166] 315.0 313.9 314.4 312.9 310.0 310.1 310.1 317.6 312.9 308.0 306.9
 [177] 305.5 297.2 293.6 300.0 292.8 288.1 285.8 286.9 283.8 193.6 191.7
 [188] 158.9 169.1 161.4 166.9 165.0 164.1 168.5 155.4 166.7 161.3 163.1
 [199] 152.5 147.0 154.6 161.0 158.9 159.3 165.4 160.9 151.5 157.2 166.1
 [210] 161.9 161.4 167.9 162.2 166.7 163.5 164.0 170.2 166.3 177.2 174.5
 [221] 168.1 189.4 188.9 176.8 181.7 268.0 261.1 258.5 261.7 264.2 258.3
 [232] 266.4 267.4 265.7 265.2 264.2 266.3 262.8 262.4 265.5 270.7 261.6
 [243] 272.9 275.8 271.0 276.3 274.0 276.1 278.1 274.5 276.6 271.6 271.9
 [254] 272.8 271.0 271.9 277.7 267.8 266.1 269.5 279.1 312.1 314.1 309.7
 [265] 310.2 302.1 301.4 304.5 305.6 312.9 306.0 308.3 305.1 296.7 299.9
 [276] 298.2 300.2 300.4 283.8 277.6 286.2 288.7 293.2 288.2 286.5 191.9
 [287] 198.2 170.3 162.5 160.2 146.1 159.8 165.4 154.7 156.3 154.1 160.8
 [298] 157.0 155.9 161.3 161.0 153.8 143.3 154.8 163.6 161.8 160.7 160.6
 [309] 157.2 154.1 162.3 161.4 156.6 161.7 164.4 178.8 184.2 178.9 184.6
 [320] 174.9 180.3 272.6 263.5 257.3 262.8 269.4 266.7 262.9 265.7 263.7
 [331] 269.7 265.9 274.2 273.6 268.6 268.4 265.5 269.2 271.1 273.5 277.8
 [342] 270.9 273.0 269.8 274.5 267.4 273.3 264.5 265.5 267.1 272.7 268.4
 [353] 271.0 268.6 266.1 270.9 279.3 312.1 308.1 305.1 302.3 305.0 305.9
 [364] 298.4 304.0 309.3 302.2 302.2 311.6 308.7 305.6 303.5 303.0 300.4
 [375] 299.3 299.8 287.9 288.7 291.7 290.3 288.6 193.7 198.5 167.3 162.3
 [386] 160.2 148.8 154.0 164.0 168.6 153.6 148.7 159.4 151.4 155.0 154.7
 [397] 158.6 156.2 149.8 155.3 159.4 160.1 155.2 161.5 159.8 157.4 161.1
 [408] 162.0 160.0 163.4 166.2 170.5 175.0 184.7 198.3 181.2 192.8 274.0
 [419] 269.9 258.8 261.8 268.8 264.5 268.5 266.6 261.6 269.2 256.9 260.2
 [430] 269.4 269.5 267.9 266.6 272.3 271.9 265.1 257.4 260.7 261.0 265.4
 [441] 274.3 273.9 265.8 269.7 264.4 271.3 267.3 258.4 265.7 272.1 268.7
 [452] 266.7 272.3 308.3 310.2 308.3 310.1 309.3 303.7 303.0 304.4 313.9
 [463] 307.8 310.6 301.8 308.0 304.7 305.8 309.5 305.5 303.7 302.5 293.2
 [474] 284.6 286.5 283.0 283.2 187.2 194.6 156.2 161.1 155.5 155.0 153.2
 [485] 159.6 164.4 151.6 143.0 147.9 155.2 157.7 152.9 158.5 146.4 145.2
 [496] 156.3 154.7 159.3 159.2 160.4 157.6 153.8 152.7 158.6 155.5 153.8
 [507] 163.4 168.4 168.0 184.1 193.5 172.9 182.9 275.5 265.7 259.1 262.5
 [518] 265.5 258.0 261.1 268.9 259.0 256.4 264.4 268.2 275.9 270.9 268.6
 [529] 264.5 263.5 270.2 269.6 274.1 274.3 270.8 274.5 274.9 273.7 276.2
 [540] 266.0 270.8 270.8 265.7 270.9 266.7 271.3 268.3 267.3 276.5 311.1
 [551] 310.5 313.6 308.9 308.5 308.1 304.5 302.4 309.7 306.4 303.1 301.2
 [562] 305.8 300.5 298.1 304.0 302.7 299.0 301.2 302.5 287.9 286.4 284.6
 [573] 287.1 194.4 202.0 167.0 164.6 161.4 156.2 162.8 161.8 166.0 153.5
 [584] 157.0 162.6 159.1 167.3 161.6 162.3 155.0 152.0 161.7 160.3 155.9
 [595] 160.3 161.3 159.1 164.8 167.0 170.9 170.4 173.8 178.8 172.0 168.1
 [606] 177.1 180.6 185.5 172.6 258.1 267.0 265.9 262.4 269.3 264.2 263.1
 [617] 265.0 259.6 262.9 270.5 267.2 268.9 266.7 268.5 271.7 271.5 274.8
 [628] 272.3 269.7 271.3 270.6 274.4 275.5 269.6 266.4 264.2 266.6 270.1
 [639] 270.1 274.5 275.4 271.3 271.8 271.1 280.2 319.5 317.1 317.7 311.3
 [650] 306.1 307.3 308.4 308.9 313.0 307.2 305.2 304.9 306.3 301.1 298.9
 [661] 299.9 296.6 295.4 303.8 299.3 282.4 283.3 279.9 285.1 191.7 190.5
 [672] 157.8 157.6 159.3 144.0 157.1 157.5 162.7 149.7 146.9 160.3 153.2
 [683] 156.7 154.7 154.7 151.9 144.3 151.7 156.7 150.6 154.9 150.0 153.6
 [694] 151.5 158.5 161.5 153.4 158.8 154.9 160.9 164.0 184.9 188.1 173.3
 [705] 172.4 268.0 263.1 260.0 260.0 260.8 255.7 264.3 267.3 261.0 260.9
 [716] 261.7 265.1 267.2 261.0 269.9 263.8 268.2 269.3 269.3 266.6 263.9
 [727] 264.2 270.4 270.2 262.8 263.3 270.3 272.2 270.4 267.1 275.2 270.5
 [738] 273.3 266.6 261.7 266.5 310.3 306.8 307.1 316.8 310.0 306.0 307.1
 [749] 308.4 309.3 307.2 306.4 308.5 304.3 301.2 300.0 303.5 304.6 292.1
 [760] 283.4 284.1 281.0 284.8 287.8 293.5 196.9 193.8 168.3 164.3 160.6
 [771] 152.8 152.6 160.8 154.4 155.4 151.9 149.4 145.5 146.3 148.8 147.8
 [782] 148.4 153.5 149.3 147.0 156.0 157.7 159.7 153.0 153.2 155.5 161.0
 [793] 160.6 158.5 167.0 161.2 162.6 175.0 193.2 180.5 180.2 261.1 256.5
 [804] 256.6 254.6 258.2 257.3 261.4 254.6 252.1 260.3 264.1 265.1 271.1
 [815] 267.5 261.3 268.3 260.2 267.4 270.7 265.2 269.5 268.9 269.4 270.8
 [826] 270.0 273.4 271.3 268.0 276.4 273.4 270.2 273.3 266.8 270.1 272.0
 [837] 272.0 310.5 314.4 314.0 312.9 300.6 307.2 303.8 307.5 313.8 306.3
 [848] 298.1 302.4 304.1 306.9 307.0 306.6 293.1 294.3 291.3 287.6 199.3
 [859] 199.4 163.0 165.6 161.7 166.4 166.8 169.9 170.3 160.4 163.9 173.1
 [870] 169.6 162.9 154.9 159.2 150.8 168.0 163.9 168.6 152.0 158.0 168.4
 [881] 169.7 164.1 163.1 167.4 168.6 166.4 173.7 172.1 175.2 171.5 183.2
 [892] 199.1 190.8 190.1 185.9 185.2 184.1 269.8 264.6 262.6 263.7 263.9
 [903] 258.2 259.2 266.0 264.5 270.7 268.3 269.4 264.3 267.7 267.0 273.4
 [914] 276.2 272.9 272.3 270.2 269.1 273.1 271.2 267.1 271.1 266.2 272.9
 [925] 269.3 265.0 261.7 267.1 264.8 267.3 263.5 264.1 263.5 307.0 311.0
 [936] 308.4 311.6 304.2 304.3 295.3 293.3 302.4 304.1 302.5 301.3 298.8
 [947] 303.2 300.3 298.3 296.6 292.7 296.8 291.8 290.1 288.7 287.3 285.6
 [958] 189.2 199.0 163.9 160.0 164.6 152.9 165.8 160.7 154.1 162.7 161.1
 [969] 159.1 160.4 161.0 161.9 163.4 160.2 156.1 160.8 160.3 164.3 164.7
 [980] 161.5 164.1 165.5 171.3 176.7 168.5 170.0 178.3 180.0 179.2 187.0
 [991] 200.6 185.5 190.1 282.3 275.1 266.5 273.4 283.3 273.8 278.7
 [ reached getOption("max.print") -- omitted 3891 entries ]

Visualisation de la Série Temporelle

autoplot(elec)

On remarque un pic à 0 dans les données. Ce sont des valeurs à remplacer pour permettre la suite des analyses.

Gestion des valeurs à 0

On remplace à vue d’oeil les valeurs nulles. En effet, on peut voir que ces valeurs sont sur un pic qui tourne autour de 150. On fixe donc ces valeurs à 150.

newdata$`Power (kW)`[newdata$`Power (kW)` == 0] <- 150 # remplacées à l'oeil nu

elec <- ts(newdata$`Power (kW)`, start=c(1,1), end=c(51,96), freq=96)
print(elec)

Time Series:
Start = c(1, 1) 
End = c(51, 96) 
Frequency = 96 
   [1] 163.1 154.4 152.2 158.7 163.8 158.7 152.3 155.2 155.9 152.1 154.1
  [12] 155.9 156.8 153.9 152.2 165.6 168.8 160.5 160.6 161.1 160.7 157.0
  [23] 161.7 158.4 165.8 166.4 168.0 159.5 164.2 170.3 178.8 181.5 182.4
  [34] 270.9 269.4 273.1 268.3 277.7 269.8 268.0 258.3 260.7 257.2 256.3
  [45] 255.0 270.9 269.6 307.6 283.5 266.1 295.9 278.5 269.5 297.5 294.4
  [56] 300.7 287.5 288.4 288.4 283.9 310.4 285.3 288.9 277.1 280.3 271.8
  [67] 301.8 282.2 285.0 326.2 311.7 313.4 305.0 300.7 303.2 299.8 307.7
  [78] 315.8 304.1 305.6 290.9 289.4 285.2 276.6 282.0 291.6 286.9 290.1
  [89] 285.1 285.7 286.6 281.9 281.3 190.3 194.8 163.7 159.3 158.4 152.9
 [100] 159.6 161.6 159.9 149.8 148.7 153.2 150.6 151.6 160.6 165.7 154.9
 [111] 149.0 163.5 161.2 158.6 154.9 162.0 162.7 160.3 162.3 168.3 165.3
 [122] 166.8 166.7 172.6 171.1 175.5 185.4 188.8 186.6 267.0 267.3 262.6
 [133] 258.2 262.0 259.4 256.6 258.2 262.8 264.7 262.9 263.8 270.7 264.2
 [144] 269.3 265.9 265.7 261.7 270.9 280.0 281.0 278.9 276.5 276.0 273.1
 [155] 270.2 276.3 265.0 266.2 267.3 275.4 273.0 273.0 272.5 266.3 284.0
 [166] 315.0 313.9 314.4 312.9 310.0 310.1 310.1 317.6 312.9 308.0 306.9
 [177] 305.5 297.2 293.6 300.0 292.8 288.1 285.8 286.9 283.8 193.6 191.7
 [188] 158.9 169.1 161.4 166.9 165.0 164.1 168.5 155.4 166.7 161.3 163.1
 [199] 152.5 147.0 154.6 161.0 158.9 159.3 165.4 160.9 151.5 157.2 166.1
 [210] 161.9 161.4 167.9 162.2 166.7 163.5 164.0 170.2 166.3 177.2 174.5
 [221] 168.1 189.4 188.9 176.8 181.7 268.0 261.1 258.5 261.7 264.2 258.3
 [232] 266.4 267.4 265.7 265.2 264.2 266.3 262.8 262.4 265.5 270.7 261.6
 [243] 272.9 275.8 271.0 276.3 274.0 276.1 278.1 274.5 276.6 271.6 271.9
 [254] 272.8 271.0 271.9 277.7 267.8 266.1 269.5 279.1 312.1 314.1 309.7
 [265] 310.2 302.1 301.4 304.5 305.6 312.9 306.0 308.3 305.1 296.7 299.9
 [276] 298.2 300.2 300.4 283.8 277.6 286.2 288.7 293.2 288.2 286.5 191.9
 [287] 198.2 170.3 162.5 160.2 146.1 159.8 165.4 154.7 156.3 154.1 160.8
 [298] 157.0 155.9 161.3 161.0 153.8 143.3 154.8 163.6 161.8 160.7 160.6
 [309] 157.2 154.1 162.3 161.4 156.6 161.7 164.4 178.8 184.2 178.9 184.6
 [320] 174.9 180.3 272.6 263.5 257.3 262.8 269.4 266.7 262.9 265.7 263.7
 [331] 269.7 265.9 274.2 273.6 268.6 268.4 265.5 269.2 271.1 273.5 277.8
 [342] 270.9 273.0 269.8 274.5 267.4 273.3 264.5 265.5 267.1 272.7 268.4
 [353] 271.0 268.6 266.1 270.9 279.3 312.1 308.1 305.1 302.3 305.0 305.9
 [364] 298.4 304.0 309.3 302.2 302.2 311.6 308.7 305.6 303.5 303.0 300.4
 [375] 299.3 299.8 287.9 288.7 291.7 290.3 288.6 193.7 198.5 167.3 162.3
 [386] 160.2 148.8 154.0 164.0 168.6 153.6 148.7 159.4 151.4 155.0 154.7
 [397] 158.6 156.2 149.8 155.3 159.4 160.1 155.2 161.5 159.8 157.4 161.1
 [408] 162.0 160.0 163.4 166.2 170.5 175.0 184.7 198.3 181.2 192.8 274.0
 [419] 269.9 258.8 261.8 268.8 264.5 268.5 266.6 261.6 269.2 256.9 260.2
 [430] 269.4 269.5 267.9 266.6 272.3 271.9 265.1 257.4 260.7 261.0 265.4
 [441] 274.3 273.9 265.8 269.7 264.4 271.3 267.3 258.4 265.7 272.1 268.7
 [452] 266.7 272.3 308.3 310.2 308.3 310.1 309.3 303.7 303.0 304.4 313.9
 [463] 307.8 310.6 301.8 308.0 304.7 305.8 309.5 305.5 303.7 302.5 293.2
 [474] 284.6 286.5 283.0 283.2 187.2 194.6 156.2 161.1 155.5 155.0 153.2
 [485] 159.6 164.4 151.6 143.0 147.9 155.2 157.7 152.9 158.5 146.4 145.2
 [496] 156.3 154.7 159.3 159.2 160.4 157.6 153.8 152.7 158.6 155.5 153.8
 [507] 163.4 168.4 168.0 184.1 193.5 172.9 182.9 275.5 265.7 259.1 262.5
 [518] 265.5 258.0 261.1 268.9 259.0 256.4 264.4 268.2 275.9 270.9 268.6
 [529] 264.5 263.5 270.2 269.6 274.1 274.3 270.8 274.5 274.9 273.7 276.2
 [540] 266.0 270.8 270.8 265.7 270.9 266.7 271.3 268.3 267.3 276.5 311.1
 [551] 310.5 313.6 308.9 308.5 308.1 304.5 302.4 309.7 306.4 303.1 301.2
 [562] 305.8 300.5 298.1 304.0 302.7 299.0 301.2 302.5 287.9 286.4 284.6
 [573] 287.1 194.4 202.0 167.0 164.6 161.4 156.2 162.8 161.8 166.0 153.5
 [584] 157.0 162.6 159.1 167.3 161.6 162.3 155.0 152.0 161.7 160.3 155.9
 [595] 160.3 161.3 159.1 164.8 167.0 170.9 170.4 173.8 178.8 172.0 168.1
 [606] 177.1 180.6 185.5 172.6 258.1 267.0 265.9 262.4 269.3 264.2 263.1
 [617] 265.0 259.6 262.9 270.5 267.2 268.9 266.7 268.5 271.7 271.5 274.8
 [628] 272.3 269.7 271.3 270.6 274.4 275.5 269.6 266.4 264.2 266.6 270.1
 [639] 270.1 274.5 275.4 271.3 271.8 271.1 280.2 319.5 317.1 317.7 311.3
 [650] 306.1 307.3 308.4 308.9 313.0 307.2 305.2 304.9 306.3 301.1 298.9
 [661] 299.9 296.6 295.4 303.8 299.3 282.4 283.3 279.9 285.1 191.7 190.5
 [672] 157.8 157.6 159.3 144.0 157.1 157.5 162.7 149.7 146.9 160.3 153.2
 [683] 156.7 154.7 154.7 151.9 144.3 151.7 156.7 150.6 154.9 150.0 153.6
 [694] 151.5 158.5 161.5 153.4 158.8 154.9 160.9 164.0 184.9 188.1 173.3
 [705] 172.4 268.0 263.1 260.0 260.0 260.8 255.7 264.3 267.3 261.0 260.9
 [716] 261.7 265.1 267.2 261.0 269.9 263.8 268.2 269.3 269.3 266.6 263.9
 [727] 264.2 270.4 270.2 262.8 263.3 270.3 272.2 270.4 267.1 275.2 270.5
 [738] 273.3 266.6 261.7 266.5 310.3 306.8 307.1 316.8 310.0 306.0 307.1
 [749] 308.4 309.3 307.2 306.4 308.5 304.3 301.2 300.0 303.5 304.6 292.1
 [760] 283.4 284.1 281.0 284.8 287.8 293.5 196.9 193.8 168.3 164.3 160.6
 [771] 152.8 152.6 160.8 154.4 155.4 151.9 149.4 145.5 146.3 148.8 147.8
 [782] 148.4 153.5 149.3 147.0 156.0 157.7 159.7 153.0 153.2 155.5 161.0
 [793] 160.6 158.5 167.0 161.2 162.6 175.0 193.2 180.5 180.2 261.1 256.5
 [804] 256.6 254.6 258.2 257.3 261.4 254.6 252.1 260.3 264.1 265.1 271.1
 [815] 267.5 261.3 268.3 260.2 267.4 270.7 265.2 269.5 268.9 269.4 270.8
 [826] 270.0 273.4 271.3 268.0 276.4 273.4 270.2 273.3 266.8 270.1 272.0
 [837] 272.0 310.5 314.4 314.0 312.9 300.6 307.2 303.8 307.5 313.8 306.3
 [848] 298.1 302.4 304.1 306.9 307.0 306.6 293.1 294.3 291.3 287.6 199.3
 [859] 199.4 163.0 165.6 161.7 166.4 166.8 169.9 170.3 160.4 163.9 173.1
 [870] 169.6 162.9 154.9 159.2 150.8 168.0 163.9 168.6 152.0 158.0 168.4
 [881] 169.7 164.1 163.1 167.4 168.6 166.4 173.7 172.1 175.2 171.5 183.2
 [892] 199.1 190.8 190.1 185.9 185.2 184.1 269.8 264.6 262.6 263.7 263.9
 [903] 258.2 259.2 266.0 264.5 270.7 268.3 269.4 264.3 267.7 267.0 273.4
 [914] 276.2 272.9 272.3 270.2 269.1 273.1 271.2 267.1 271.1 266.2 272.9
 [925] 269.3 265.0 261.7 267.1 264.8 267.3 263.5 264.1 263.5 307.0 311.0
 [936] 308.4 311.6 304.2 304.3 295.3 293.3 302.4 304.1 302.5 301.3 298.8
 [947] 303.2 300.3 298.3 296.6 292.7 296.8 291.8 290.1 288.7 287.3 285.6
 [958] 189.2 199.0 163.9 160.0 164.6 152.9 165.8 160.7 154.1 162.7 161.1
 [969] 159.1 160.4 161.0 161.9 163.4 160.2 156.1 160.8 160.3 164.3 164.7
 [980] 161.5 164.1 165.5 171.3 176.7 168.5 170.0 178.3 180.0 179.2 187.0
 [991] 200.6 185.5 190.1 282.3 275.1 266.5 273.4 283.3 273.8 278.7
 [ reached getOption("max.print") -- omitted 3896 entries ]

autoplot(elec) # visualisation des valeurs remplacées

Décomposition de la Série Temporelle

On décompose la série.

autoplot(decompose(elec, type="additive"))

Il ne semble pas y avoir de tendance. On se concentrera donc sur des modèles saisonniers.

Division des Données

On divise nos données en ensembles d’apprentissage et de test pour un rapport de 80/20. L’ensemble d’apprentissage commence ainsi le premier jour du jeu de données (2 Janvier 2010) à la première heure, et se termine le quarantième jour (10 Février 2010) à la dernière heure.

train <- window(elec, start=c(1,1), end=c(40,96))
test <- window(elec, start=c(41,1), end=c(50,96))

On affiche les 2 ensembles simultanément :

plot(train, xlim=c(1,52), ylim=c(100,380))
lines(test, lty=2)

Lissage Exponentiel Simple

On lance un Lissage Exponentiel Simple, car la meilleure prédiction a priori est une constante.

Modélisation :

LES = HoltWinters(train, alpha=NULL, beta=FALSE, gamma=FALSE)
print(LES)

Holt-Winters exponential smoothing without trend and without seasonal component.

Call:
HoltWinters(x = train, alpha = NULL, beta = FALSE, gamma = FALSE)

Smoothing parameters:
 alpha: 0.9860426
 beta : FALSE
 gamma: FALSE

Coefficients:
      [,1]
a 156.3719

On prédit avec la constante trouvée précédemment. On constate que la moyenne, qui équivaut à la moyenne des valeurs de ‘Power’ est aux environs de 150. Notre valeur fixée a priori pour remplacer les valeurs nulles n’est pas absurde, bien que l’on pourrait la remplacer par la valeur exacte calculée par ce modèle.

Prédiction :

pred1 <- predict(LES, n.ahead=960) # prédiction sur les 10 jours suivants
plot(test)
lines(pred1, col=2) # prédiction à partir du train set

Coefficient de la constante du Lissage Exponentiel Simple :

print(LES$alpha) # 0.9860426

[1] 0.9860426

Évaluation - RMSE du Lissage Exponentiel Simple :

print(sqrt(mean((pred1-test)^2))) # 93.28624

[1] 93.28624

Holt Winters Saisonnier

On lance un Holt Winters saisonnier avec une constante alpha et une saisonnalité gamma.

Modélisation :

HW = HoltWinters(train, alpha=NULL, beta=FALSE, gamma=NULL)
print(HW)

Holt-Winters exponential smoothing without trend and with additive seasonal component.

Call:
HoltWinters(x = train, alpha = NULL, beta = FALSE, gamma = NULL)

Smoothing parameters:
 alpha: 0.7831196
 beta : FALSE
 gamma: 0.8904545

Coefficients:
         [,1]
a   240.11894
s1  -85.53653
s2  -81.59209
s3  -84.89480
s4  -78.20116
s5  -72.88999
s6  -71.68576
s7  -85.21335
s8  -83.36854
s9  -79.97476
s10 -79.19373
s11 -78.56924
s12 -83.33520
s13 -85.83806
s14 -89.88809
s15 -88.66898
s16 -83.77997
s17 -81.77421
s18 -82.99303
s19 -81.97842
s20 -79.56752
s21 -78.70675
s22 -78.71416
s23 -80.33166
s24 -76.59747
s25 -75.18980
s26 -68.89655
s27 -57.44692
s28 -56.74519
s29 -60.28571
s30 -61.69323
s31 -70.64407
s32 -63.30628
s33 -58.80193
s34  23.24709
s35  24.12167
s36  20.54998
s37  22.51192
s38  28.38224
s39  22.49509
s40  23.29417
s41  26.44203
s42  22.38805
s43  24.39604
s44  26.29052
s45  24.03414
s46  25.34776
s47  24.20117
s48  22.40583
s49  24.49166
s50  27.72696
s51  35.45720
s52  37.98888
s53  30.40060
s54  34.20281
s55  35.84587
s56  35.46848
s57  35.39524
s58  38.46045
s59  41.43965
s60  42.53769
s61  41.54371
s62  44.69122
s63  41.75707
s64  46.65363
s65  45.10486
s66  47.29578
s67  49.76163
s68  48.48894
s69  43.75145
s70  47.81166
s71  52.60843
s72 105.88779
s73 108.53680
s74 103.49910
s75  95.49748
s76  89.73647
s77  84.74433
s78  83.48915
s79  74.42472
s80  73.32390
s81  70.05516
s82  67.09232
s83  60.54797
s84  60.42753
s85  59.10355
s86  55.27885
s87  52.05517
s88  51.74057
s89  47.41802
s90  29.00408
s91  25.76486
s92  18.88309
s93  17.50113
s94 -57.21982
s95 -55.69980
s96 -84.05392

On voit bien les 96 périodes de la saisonnalité.

Prédiction :

pred2 <- predict(HW, n.ahead=960)
plot(test)
lines(pred2, col=3) # prédiction à partir du train set

A priori, ce modèle n’est pas si mauvais.

Coefficients de la constante et de la saisonnalité du Holt Winters :

print(HW$alpha) # 0.7831196

    alpha 
0.7831196 

print(HW$gamma) # 0.8904545

    gamma 
0.8904545 

Évaluation - RMSE du Holt Winters saisonnier :

print(sqrt(mean((pred2-test)^2))) # 21.26563

[1] 21.26563

Auto-ARIMA

Après avoir lancé les modèles a priori, on continue avec un auto-ARIMA pour trouver le meilleur modèle théorique.

Modélisation :

model3 = auto.arima(train)
install.packages("openxlsx")

trying URL 'https://cran.rstudio.com/bin/macosx/big-sur-arm64/contrib/4.3/openxlsx_4.2.5.2.tgz'
Content type 'application/x-gzip' length 3246918 bytes (3.1 MB)
==================================================
downloaded 3.1 MB

The downloaded binary packages are in
    /var/folders/mk/hc7kkglj5fncc3271_yfcf6r0000gn/T//RtmpUrPkjh/downloaded_packages

summary(model3) # AIC : 27696.77

Series: train 
ARIMA(1,0,0)(0,1,0)[96] 

Coefficients:
         ar1
      0.7815
s.e.  0.0102

sigma^2 = 95.46:  log likelihood = -13846.39
AIC=27696.77   AICc=27696.77   BIC=27709.23

Training set error measures:
                      ME     RMSE      MAE        MPE     MAPE      MASE
Training set -0.06254446 9.646271 5.620404 -0.1185649 2.611593 0.7134771
                     ACF1
Training set 0.0003086627

L’auto-ARIMA nous donne un SARIMA d’ordre 1, , et de période 96.

Prédiction :

pred3 = forecast(model3, h=960)

RMSE de l’ARIMA :

print(sqrt(mean((pred3$mean-test)^2))) # 15.71738

[1] 15.71738

Notre modèle généré par l’auto-ARIMA est le meilleur jusqu’ici des 3 créés. Mais nous devons maintenant nous assurer que les résidus de la série sont indépendants du passé.

checkresiduals(model3)

    Ljung-Box test

data:  Residuals from ARIMA(1,0,0)(0,1,0)[96]
Q* = 1522.8, df = 191, p-value < 2.2e-16

Model df: 1.   Total lags used: 192

Les résidus ne sont donc pas indépendants. Nous devons par conséquent différencier la série afin d’extraire les résidus et les rendre indépendants.

Suppression Saisonnalité

Il faudrait supprimer la saisonnalité de la série temporelle afin de pouvoir lancer un SARIMA.Pour cela, il faudrait différencier la série avec un lag spécifique. Il faudrait également calculer les auto-corrélations (ggAcf) et les auto-corrélations partielles (ggPacf) pour trouver l’ordre du SARIMA. Il faudrait également s’assurer de l’indépendance des résidus avec un box-test : les résidus sont-ils du bruit blanc ?

tmp = diff(train, lag=96)
plot(tmp)

ggAcf(tmp)

ggPacf(tmp)

On constate que les résidus ne sont pas indépendants après une différenciation de la série.

tmp1 = diff(tmp, lag=192)
plot(tmp1)

ggAcf(tmp1)

ggPacf(tmp1)

Après avoir rendu les résidus de la série indépendants, nous pouvons lancer un SARIMA avec les paramètres trouvés précédemment, ainsi que d’autres modèles.

Réseaux de Neurones

Modélisation :

model4 = nnetar(train)
print(model4)

Series: train 
Model:  NNAR(20,1,11)[96] 
Call:   nnetar(y = train)

Average of 20 networks, each of which is
a 21-11-1 network with 254 weights
options were - linear output units 

sigma^2 estimated as 46.92

Prédiction :

pred4 = forecast(model4, h=960)

Évaluation :

print(sqrt(mean((pred4$mean-test)^2)))

[1] 68.82039

Les réseaux de neurones ne sont pas très bon, probablement parce que notre série n’est pas stationnaire.

Comparaison de Modèles

Nous insérons un graphique qui affiche simultanément chaque prédiction.

par(mfrow=c(1,1))

plot(test, xlim=c(41,51), ylim=c(120,700))
lines(test, lty=2)

lines(pred1, col=2)
lines(pred2, col=3)

lines(pred3$mean, col=4)
lines(pred4$mean, col=5)

legend('topleft', 
       col=1:5, 
       lty=1, 
       legend=c('Vraies Données', 
                'Prédictions avec LES', 
                'Prédictions avec HW', 
                'ARIMA', 
                'Réseaux de Neurones'))

Prédiction avec le Meilleur Modèle

SAR = Arima(elec, order=c(1,0,0), seasonal=c(0,1,0))
summary(SAR)

Series: elec 
ARIMA(1,0,0)(0,1,0)[96] 

Coefficients:
         ar1
      0.7164
s.e.  0.0102

sigma^2 = 128.3:  log likelihood = -18092.79
AIC=36189.58   AICc=36189.58   BIC=36202.49

Training set error measures:
                     ME     RMSE      MAE        MPE     MAPE      MASE
Training set -0.1144681 11.21379 6.400623 -0.1672996 2.887978 0.7451869
                   ACF1
Training set -0.0708581

pred = forecast(SAR, h=96)

autoplot(elec)+autolayer(pred)

#predictions <- as.numeric(pred$mean)
#pred_df <- data.frame(Prediction = predictions)

Séries Multivariées

Ici, nous tentons de prédire la consommation d’électricité avec la température comme covariable.

power <- ts(newdata$`Power (kW)`, start=c(1,6), end=c(51,96), freq=96)
temperature <- ts(newdata$`Temp (C°)`, start=c(1,6), end=c(51,96), freq=96)

power_train <- window(power, start=c(1,1), end=c(40,96))
power_test <- window(power, start=c(41,1), end=c(50,96))

temperature_train <- window(temperature, start=c(1,1), end=c(40,96))
temperature_test <- window(temperature, start=c(41,1), end=c(50,96))

fit1 = tslm(power_train ~ temperature_train)
summary(fit1)

Call:
tslm(formula = power_train ~ temperature_train)

Residuals:
     Min       1Q   Median       3Q      Max 
-121.545  -42.092    2.155   43.037  111.973 

Coefficients:
                  Estimate Std. Error t value Pr(>|t|)    
(Intercept)       126.0818     3.4186   36.88   <2e-16 ***
temperature_train   9.9478     0.3137   31.71   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 50.83 on 3833 degrees of freedom
Multiple R-squared:  0.2078,    Adjusted R-squared:  0.2076 
F-statistic:  1005 on 1 and 3833 DF,  p-value: < 2.2e-16

fit2 = tslm(power_train ~ temperature_train+season+trend)
summary(fit2)

Call:
tslm(formula = power_train ~ temperature_train + season + trend)

Residuals:
     Min       1Q   Median       3Q      Max 
-112.032   -4.617    0.248    4.591   58.331 

Coefficients:
                    Estimate Std. Error t value Pr(>|t|)    
(Intercept)        2.611e+02  2.092e+00 124.776  < 2e-16 ***
temperature_train  1.039e+00  9.814e-02  10.589  < 2e-16 ***
season2            1.107e+00  2.622e+00   0.422  0.67291    
season3           -7.446e+01  2.623e+00 -28.392  < 2e-16 ***
season4           -7.221e+01  2.623e+00 -27.533  < 2e-16 ***
season5           -1.013e+02  2.623e+00 -38.632  < 2e-16 ***
season6           -1.044e+02  2.606e+00 -40.074  < 2e-16 ***
season7           -1.050e+02  2.607e+00 -40.273  < 2e-16 ***
season8           -1.111e+02  2.607e+00 -42.637  < 2e-16 ***
season9           -1.048e+02  2.607e+00 -40.206  < 2e-16 ***
season10          -1.016e+02  2.607e+00 -38.978  < 2e-16 ***
season11          -1.012e+02  2.607e+00 -38.801  < 2e-16 ***
season12          -1.108e+02  2.607e+00 -42.478  < 2e-16 ***
season13          -1.109e+02  2.607e+00 -42.526  < 2e-16 ***
season14          -1.075e+02  2.607e+00 -41.237  < 2e-16 ***
season15          -1.083e+02  2.608e+00 -41.515  < 2e-16 ***
season16          -1.055e+02  2.608e+00 -40.448  < 2e-16 ***
season17          -1.063e+02  2.608e+00 -40.754  < 2e-16 ***
season18          -1.048e+02  2.608e+00 -40.180  < 2e-16 ***
season19          -1.093e+02  2.609e+00 -41.915  < 2e-16 ***
season20          -1.107e+02  2.609e+00 -42.453  < 2e-16 ***
season21          -1.052e+02  2.609e+00 -40.317  < 2e-16 ***
season22          -1.036e+02  2.609e+00 -39.729  < 2e-16 ***
season23          -1.046e+02  2.610e+00 -40.084  < 2e-16 ***
season24          -1.056e+02  2.610e+00 -40.447  < 2e-16 ***
season25          -1.039e+02  2.610e+00 -39.824  < 2e-16 ***
season26          -1.041e+02  2.610e+00 -39.901  < 2e-16 ***
season27          -1.032e+02  2.611e+00 -39.523  < 2e-16 ***
season28          -1.022e+02  2.611e+00 -39.139  < 2e-16 ***
season29          -1.002e+02  2.611e+00 -38.380  < 2e-16 ***
season30          -1.008e+02  2.611e+00 -38.625  < 2e-16 ***
season31          -9.727e+01  2.611e+00 -37.255  < 2e-16 ***
season32          -9.022e+01  2.611e+00 -34.552  < 2e-16 ***
season33          -8.799e+01  2.611e+00 -33.698  < 2e-16 ***
season34          -8.752e+01  2.611e+00 -33.520  < 2e-16 ***
season35          -8.208e+01  2.611e+00 -31.435  < 2e-16 ***
season36          -8.259e+01  2.611e+00 -31.628  < 2e-16 ***
season37          -8.801e+01  2.611e+00 -33.707  < 2e-16 ***
season38          -8.436e+01  2.611e+00 -32.309  < 2e-16 ***
season39          -3.692e-01  2.611e+00  -0.141  0.88754    
season40          -2.332e+00  2.611e+00  -0.893  0.37177    
season41          -5.300e+00  2.611e+00  -2.030  0.04242 *  
season42          -5.585e+00  2.611e+00  -2.139  0.03247 *  
season43          -2.926e+00  2.606e+00  -1.123  0.26159    
season44          -7.947e+00  2.606e+00  -3.049  0.00231 ** 
season45          -5.467e+00  2.606e+00  -2.098  0.03600 *  
season46          -4.552e+00  2.606e+00  -1.747  0.08077 .  
season47          -8.393e+00  2.607e+00  -3.220  0.00129 ** 
season48          -7.586e+00  2.607e+00  -2.910  0.00364 ** 
season49          -5.989e+00  2.607e+00  -2.297  0.02166 *  
season50          -5.814e+00  2.607e+00  -2.230  0.02579 *  
season51          -3.773e+00  2.611e+00  -1.445  0.14846    
season52          -5.596e+00  2.611e+00  -2.144  0.03214 *  
season53          -5.069e+00  2.611e+00  -1.942  0.05226 .  
season54          -4.369e+00  2.611e+00  -1.674  0.09429 .  
season55          -4.660e+00  2.618e+00  -1.780  0.07508 .  
season56          -1.243e+00  2.618e+00  -0.475  0.63483    
season57          -2.054e+00  2.617e+00  -0.785  0.43275    
season58          -3.919e+00  2.617e+00  -1.497  0.13442    
season59          -2.328e+00  2.623e+00  -0.888  0.37486    
season60          -1.916e+00  2.623e+00  -0.730  0.46520    
season61          -2.651e+00  2.623e+00  -1.011  0.31220    
season62          -3.351e+00  2.623e+00  -1.278  0.20140    
season63          -3.150e+00  2.628e+00  -1.199  0.23062    
season64          -2.996e+00  2.628e+00  -1.140  0.25431    
season65          -3.221e+00  2.628e+00  -1.226  0.22032    
season66          -3.431e+00  2.628e+00  -1.306  0.19165    
season67          -3.417e+00  2.628e+00  -1.300  0.19364    
season68          -4.117e+00  2.628e+00  -1.567  0.11729    
season69          -4.015e+00  2.628e+00  -1.528  0.12666    
season70          -4.161e+00  2.628e+00  -1.583  0.11349    
season71          -3.996e+00  2.623e+00  -1.523  0.12781    
season72          -3.411e+00  2.623e+00  -1.300  0.19359    
season73          -5.951e+00  2.623e+00  -2.269  0.02335 *  
season74          -6.469e+00  2.623e+00  -2.466  0.01370 *  
season75           1.426e+01  2.616e+00   5.450 5.35e-08 ***
season76           2.826e+01  2.616e+00  10.803  < 2e-16 ***
season77           4.055e+01  2.616e+00  15.504  < 2e-16 ***
season78           4.002e+01  2.616e+00  15.299  < 2e-16 ***
season79           3.773e+01  2.610e+00  14.457  < 2e-16 ***
season80           3.584e+01  2.610e+00  13.733  < 2e-16 ***
season81           3.467e+01  2.610e+00  13.284  < 2e-16 ***
season82           3.599e+01  2.610e+00  13.789  < 2e-16 ***
season83           4.104e+01  2.608e+00  15.734  < 2e-16 ***
season84           3.676e+01  2.608e+00  14.094  < 2e-16 ***
season85           3.645e+01  2.608e+00  13.974  < 2e-16 ***
season86           3.445e+01  2.608e+00  13.210  < 2e-16 ***
season87           3.413e+01  2.607e+00  13.091  < 2e-16 ***
season88           3.258e+01  2.607e+00  12.497  < 2e-16 ***
season89           3.196e+01  2.607e+00  12.259  < 2e-16 ***
season90           3.130e+01  2.607e+00  12.006  < 2e-16 ***
season91           2.991e+01  2.606e+00  11.476  < 2e-16 ***
season92           2.785e+01  2.606e+00  10.687  < 2e-16 ***
season93           2.742e+01  2.606e+00  10.520  < 2e-16 ***
season94           2.546e+01  2.606e+00   9.768  < 2e-16 ***
season95           7.612e+00  2.606e+00   2.921  0.00351 ** 
season96           6.379e+00  2.606e+00   2.448  0.01441 *  
trend             -4.632e-03  1.737e-04 -26.664  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 11.58 on 3737 degrees of freedom
Multiple R-squared:  0.9599,    Adjusted R-squared:  0.9589 
F-statistic: 922.4 on 97 and 3737 DF,  p-value: < 2.2e-16

CV(fit1)

          CV          AIC         AICc          BIC        AdjR2 
2.584402e+03 3.013516e+04 3.013516e+04 3.015391e+04 2.075976e-01 

CV(fit2) # BIC meilleur

          CV          AIC         AICc          BIC        AdjR2 
1.376823e+02 1.888504e+04 1.889034e+04 1.950398e+04 9.588661e-01 

checkresiduals(fit2, test=FALSE, plot=TRUE)

checkresiduals(fit2, test='LB', plot=FALSE)

    Ljung-Box test

data:  Residuals from Linear regression model
Q* = 8638.4, df = 192, p-value < 2.2e-16

Model df: 0.   Total lags used: 192

ggAcf(fit2$residuals)

ggPacf(fit2$residuals)

fit3 = Arima(power_train, xreg=temperature_train, order=c(1, 0, 0), seasonal=c(0, 1, 0))
summary(fit3)

Series: power_train 
Regression with ARIMA(1,0,0)(0,1,0)[96] errors 

Coefficients:
         ar1    xreg
      0.7798  0.3065
s.e.  0.0103  0.2406

sigma^2 = 95.55:  log likelihood = -13829.12
AIC=27664.24   AICc=27664.24   BIC=27682.92

Training set error measures:
                      ME     RMSE      MAE        MPE     MAPE      MASE
Training set -0.05764918 9.649274 5.619845 -0.1166433 2.610719 0.7128255
                     ACF1
Training set 0.0006469121

checkresiduals(fit3, test=FALSE)

checkresiduals(fit3, plot=FALSE)

    Ljung-Box test

data:  Residuals from Regression with ARIMA(1,0,0)(0,1,0)[96] errors
Q* = 1522.2, df = 191, p-value < 2.2e-16

Model df: 1.   Total lags used: 192

fit4 = nnetar(power_train, xreg=temperature_train)
print(fit4)

Series: power_train 
Model:  NNAR(20,1,12)[96] 
Call:   nnetar(y = power_train, xreg = temperature_train)

Average of 20 networks, each of which is
a 22-12-1 network with 289 weights
options were - linear output units 

sigma^2 estimated as 46.35

autoplot(forecast(train))